ABSTRACT

CIRCULAR POLARIZATION SPECTROSCOPY: DISORIENTATION 133 2 CROSS-SECTION IN THE Cs 6p P3/2 LEVEL BY USING TWO-PHOTON TWO-COLOR NANO-SECOND PULSED LASER

by Ramesh Marhatta

We have experimentally investigated the disorientation cross-section of the cesium J = 3/2 excited-level atom by using two-photon two-color nano-second pulsed dye lasers in the presence of argon buffer gas. Alignment and orientation 2 in the 6p P3/2 level were produced with a circularly polarized light. The circular polarization degree was measured with the pump (λ = 852.112 nm) and probe 2 laser (λ = 603.409 nm) to obtain disorientation cross-section in the 6p P3/2 level cesium due to collision with the ground level argon atoms over the Zeeman population. The disorientation cross-section was extracted, using non-linear square fit, from the measured circular polarization spectra as a function of argon gas pressure. We obtained the value of disorientation cross-section as 151(42) A˚2 which matches with the theory. CIRCULAR POLARIZATION SPECTROSCOPY: 133 2 DISORIENTATION CROSS-SECTION IN THE Cs 6p P3/2 LEVEL BY USING TWO-PHOTON TWO-COLOR NANO-SECOND PULSED LASER

A Thesis

Submitted to the

Faculty of Miami University

in partial fulfillment of

the requirements for the degree of

Master of Science

Department of Physics

by

Ramesh Marhatta

Miami University

Oxford, Ohio

2007

Advisor Bur¸cinS. Bayram

Reader Douglas Marcum

Reader Samir Bali Contents

1 Introduction 1

2 Properties of Cesium 3

3 Theory of Polarization Spectra 7 3.1 Excitation Scheme ...... 7 3.2 Selection Rules ...... 8 3.3 Alignment and Orientation ...... 9 3.4 Density Matrix Formulation ...... 11 3.5 Circular Polarization Spectra ...... 13 3.5.1 Linear and Circular Polarization of Light ...... 13 3.5.2 Intensity and Circular Polarization Degree ...... 16 3.6 Fine and Hyperfine Interactions ...... 18 3.6.1 Fine Structure ...... 18 3.6.2 Hyperfine Structure ...... 21 3.7 Hyperfine Depolarization Effects on Circular Polarization Spectra 22

4 Experimental Apparatus 25 4.1 Lasers ...... 25 4.1.1 Nd:YAG Laser ...... 25 4.1.2 Dye Laser ...... 25 4.2 Free Spectral Range ...... 29 4.3 Circular Polarization Set-up ...... 29 4.4 Cesium Cell ...... 31

5 Overview of Experimental Measurement 33

6 Signal Detection Scheme 36

7 Systematic Effects 38 7.1 Effect of Power of Laser ...... 39 7.2 Effect of Temperature ...... 40

ii 8 Rate Equation Analysis and Results 41 8.1 Rate Equation Analysis ...... 41 8.2 Polarization as a Function of Buffer Gas Pressure ...... 45

9 Conclusion and Future Directions 48

A Production and Detection of Linear and Circularly Polarized light Using Jone’s Matrices 51

B Calculation of Circular Polarization Degree 54

C LabVIEW Program 57

D Apparatus 60

iii List of Tables

2.1 Properties of naturally occurring 133Cs atoms...... 5

3.1 Table for coupling constants used in the calculations...... 22

5.1 Some important parameters involved in the experiment...... 35

8.1 Our work on the circular depolarization with argon buffer gas in 2 2 the 6s S1/2 → 10s S1/2 transition...... 46 8.2 The alignment and orientation depolarization cross-section of the 133 2 Cs 6p P3/2 J = 3/2 atoms...... 46 8.3 Some important parameters used in the experiment ...... 47

133 2 2 B.1 The alignment and orientation in the Cs 6p P1/2 and 6p P3/2 states...... 56 B.2 Clebsch-Gordan coefficients used in the experiment ...... 56

iv List of Figures

2.1 Cesium D1 and D2 lines...... 4 2.2 Vapor pressure and number density of cesium atoms...... 6

3.1 Excitation of atoms...... 7 3.2 Grotrian diagram showing the excitation and emission scheme. . . 8 3.3 Selection rule...... 9 3.4 Aligned axially symmetric system...... 10 3.5 Oriented axially symmetric system...... 11 3.6 Collision and detection frame...... 12 3.7 Transition of electrons from s to p level...... 13 3.8 Combination of two linearly polarized waves to form a resultant linearly polarized wave...... 14 3.9 Combination of two linearly polarized waves to form circularly po- larized wave...... 15 3.10 Circular polarization of light...... 15 3.11 Energy level diagram of Cs...... 17 3.12 Hydrogen atom from the electron’s perspective...... 19 3.13 Hyperfine splitting of the levels of 133Cs...... 21 3.14 Overlap time of the pump and probe lasers...... 23

4.1 Experimental apparatus...... 26 4.2 Pumping scheme of experiment...... 27 4.3 Littman-Metcalf cavity configuration...... 28 4.4 Cesium cell...... 31 4.5 View of the oven wrapped with heating tape and aluminum foil. 32

5.1 Micrometer tuning curve of laser 1...... 33 5.2 Geometry of the experiment ...... 34

7.1 Effect of power of laser 1 on circular polarization degree...... 39 7.2 Effect of power of laser 2 on circular polarization degree...... 40 7.3 Effect of temperature of cell on the circular polarization degree. . 40

8.1 Population mixing among the Zeeman coherence ...... 42

v 8.2 Weighted non-linear least square fit of the circular polarization spectrum of Cs...... 46

B.1 Transition of electrons by right circularly polarized light...... 54

C.1 Interface of the LCVR...... 57 C.2 Medowlark LCR subVI...... 58 C.3 takedata3 sub.vi...... 58 ¯ C.4 Stepper motor subVI...... 59 C.5 Get Spectrum program to take wave form of fluorescence signal. . 59

D.1 Experimental apparatus...... 60 D.2 Flowing dye laser 1 oscillator in the Littman-Metcalf design. . . . 61 D.3 Dye flowing machine for laser 1...... 61 D.4 Static dye laser 2 oscillator in Littman-Metcalf cavity...... 62 D.5 Dye laser 2 amplifier...... 62 D.6 A view of quarter wave plate, Glan-Thompson polarizer and the LCR ...... 63 D.7 Boxcar averager/integrator...... 63 D.8 DAQ board connected between a computer and the boxcar aver- ager/integrator...... 64

vi ACKNOWLEDGMENTS

It is the greatest moment to express my sincere thanks to my research advisor Dr. Bur¸cinS. Bayram for her continuous support, encouragement and guidance throughout my research. Her readiness and availability at all times to answer my questions are worthy to praise. She created a favorable environment for me to learn very basic knowledge in research. This paved the path to my brighter future. Also, I would like to appreciate the work of graduate student Seda Kin and undergraduate assistants Jacob Hinkle and Morgan Welsh who contributed a lot in this research. It is my pleasure to thank the machinist Michael Eldridge for his support in this research. I would like to appreciate my thesis committee members Dr. Samir Bali and Dr. S. Douglas Marcum for reading the manuscript and giving me valuable suggestions.

I would also like to acknowledge the Department of Physics at Miami University, faculties, and staffs for offering me a great opportunity and support to undertake this project. I cannot simply say thanks to my wife Ranju and daughter Rajshree for their inspiration throughout my study at Miami University.

Finally, I wish lifetime happiness to the Bayram’s family.

vii Chapter 1

Introduction

Lasers have wide applications in various fields of science and technology. With the invention of solid-state laser, gas laser, and dye laser, new areas of research are open to scientists. Scientists have been taking advantages using lasers in research because lasers are coherent, monochromatic and travel a long distance without much loss of energy and without divergence. Scientists are interested in the study of photon-matter interaction to under- stand the properties and behavior of the atoms and molecules. Laser light sources have been used in the study of interaction of photon with matter, photoionization or photodissociation, stimulated Raman spectroscopy, polarization spectroscopy, atoms trapping and cooling, electromagnetic induced transparency [1, 2] and so on. Nowadays lasers open up new possibilities for research. Since dye laser is tunable, it is used to study single photon, two-photon resonance (this research) and multi-photon excitation to study the lifetime of different atomic levels, decay rate and atomic ionization [3]. Two-photon excitation process has been mainly used to study the hyperfine structure [4] of atoms and quantum beat spectroscopy in the resolved hyperfine levels. In our research we use circularly polarized pulsed laser to study the collisional dynamics of the excited state of 133Cs atom colliding with ground level argon atoms. The main idea of this research is to measure the circular polarization de- 2 2 2 133 gree of 6s S1/2 → 6p P3/2 → 10s S1/2 transition in Cs atom and to extract the disalignment and disorientation cross-section. We investigated circular polariza- 133 2 2 2 tion spectrum of the Cs 6s S1/2 → 6p P3/2 → 10s S1/2 transition and measured polarization degree through two-photon double resonance with argon buffer gas at different pressures ranging from 5 to 100 Torr. We can gain valuable information on relaxation rates of electronic moments and the state-multipole-dependent depolarization cross-section by studying the depolarization in excited level alkali atoms under the influence of collision with noble gases. The study of collision between alkali atoms and noble gas is important for numerous applications, particularly for remotely sensing the composition of planetary atmospheres, the interstellar medium, and fusion plasmas [5].

1 The study of collisional dynamics is not new, but recent advances in tech- nique for cooling and trapping of atoms has enormously increased the interests in this area. For example, high-resolution molecular spectroscopy of colliding cold atoms has become a leading experimental technique to investigate the collisional dynamics and to accurately determine the potential curves of molecules. In the past different attempts were made to study depolarization cross-section of alkali atoms by using argon buffer gas [6-9]. Fricke and Luscher [6] reported collisional 133 2 depolarization cross-section in Cs 6p P3/2 level by using Cs-lamp and introduc- ing different noble gases with pressure up to 30 Torr. They used a weak magnetic field and applied different pressures to restrict the collisional frequency between Cs and argon in order to neglect the effect of I-J coupling in their polarization measurement. In 1975, Guiry et.al [7] used both linear and circularly polarized light to study the collisional depolarization cross-section of cesium atom in the 2 6p P3/2 level by using different pressures of buffer gas ranging from 0 to 10 Torr and high magnetic field (10 kG). He extracted the disorientation and disalign- 2 133 ment cross-section of 6p P3/2 Cs atoms. The Hanle effect was used to decouple J and I and to neglect hyperfine coupling. Also, Okunevich [10] and Rebane [11] reported theoretical disalignment cross-section of Cs atom in the excited state in the vicinity of various noble gases. We used pulsed laser instead of incoherent Cs-lamp and used two-photon two- color pump-probe technique between excited Cs atom and ground state argon 133 2 buffer gas to extract orientation depolarization cross-section of Cs 6p P3/2. By using pump-probe technique, we can have control over the overlapping time of the pump and probe lasers. This gives us an advantage to measure the amount of hyperfine depolarization effect to the measured signal.

2 Chapter 2

Properties of Cesium

Cesium is a beautiful alkali metal with pale golden color which sets it apart from the other alkali metals. Only the single isotope of 133Cs is stable and naturally occurring. Melting point of cesium is 28.4 0C which is the second lowest of all metals. It can easily be liquefied within the sealed vacuum cell by the warmth of one’s hand. Cesium is the most reactive of all metals and burst into flames instantly when it comes in contact with air at room temperature. It reacts with ice at temperature above -165 0C. Cesium hydroxide is a strong base and attacks the glass. It oxidizes rapidly when exposed to air and forms a dangerous super oxide on its surface. Also, it reacts explosively with water to form base which attacks glassware even at room temperature. Thus, cesium should be handled carefully. However, cesium is quite safe and stable with a small amount inside vacuum sealed cell and heated to a moderate temperature. Cesium falls on 1A of the periodic table and has an atomic number 55 and atomic mass 133. The electronic configuration of the 133Cs is very simple with 2 only one valence electron in the ground 6s S1/2 state. This valence electron moves in spherically symmetric potential described by the central field approximation. Thus the energy states of the cesium atoms obey famous Rydberg formula Enl = ∗2 ∗ -R/n , where n = n-δl is the effective principle quantum number and δl is the quantum defect. Nuclear spin for 133Cs is 7/2. Cesium has the following energy 2 level scheme. The ground 6s S1/2 state hyperfine splits into two levels given by F = I ± 1/2. The lowest excited P1/2 state has two fine splitting given by I ± 1/2. 2 The excited 6p P1/2 state has four fine splitting given by F = I + 3/2 ...... I - 3/2 2 and the 10s S1/2 state has two fine splitting given by F = I ± 3/2. The transition 2 from the ground state to lower 6p P1/2 state is called D1 line and the transition 2 to higher 6p P3/2 state is called D2 line. Generally the nuclear spin increases with atomic number of alkali atoms. The lowest excited state is closer to ground state resulting in longer wavelength (near infrared) for D lines in cesium. The fine structure splitting between D1 and D2 lines increases rapidly with atomic number, from only 10 GHz in 6Li to over 16 THz in 133Cs. The hyperfine splitting in both ground and excited states also increases with atomic number from about

3 228 MHz in lithium to 9192.631770 MHz in 133Cs. Larger splitting means ground state and D1 excited state hyperfine component can be resolved even at room temperature. The wavelengths of D1 and D2 lines in air are 894.353 nm and 852.118 nm respectively [12]. By using the fast beam laser technique, the lifetime 2 2 for the 6p P1/2 state is calculated to be 35.07(10) ns and 30.57(7) ns for the 6p P3/2 state. Cesium D1 and D2 lines are shown in the Figure 2.1.

Figure 2.1: Cesium D1 and D2 lines.

As the atomic number increases the increasing shielding of the nuclear charges by the additional closed electronic shell results in increasing electropositivity and decreasing ionization potential energy. The ionization potential energy for Cs is 31406.46766(15) cm−1 [12]. Some of the properties of the Cs atoms are listed in Table 2.1. Since the cesium has only one valence electron, it is hydrogen-like atom so it gives relative ease to formulate the atomic theory calculations. Cesium is mainly used in atom trapping and cooling [13] and atomic clock development [14] and more recently in ion propulsion system. Since the melting point of Cs is very low we can make cesium vapor even at room temperature. The vapor pressure and atomic density depend on the temperature of the system. The vapor density of cesium atoms ranges from 1010 atoms/cm3 to 1015 atoms/cm3 at the temperature range of 25 0C to 200 0C. The vapor pressure rises nearly exponentially with the rise in temperature and can be calculated from the Equations 2.1 and 2.2 [12]. For solid phase, 1088.676 log P = −219.482 00 + − 0.083 361 85 T + 94.887 52 log T (2.1) 10 v T 10

4 Table 2.1: Properties of naturally occurring 133Cs atoms.

Natural Abundance 100.0 % Nuclear spin 7/2 λD1 894.6 nm λD2 852.3 nm Fine Structure Splitting (D1-D2) 16623 GHz 2 Hyperfine splitting S1/2 9193 MHz 2 Hyperfine splitting P1/2 1168 MHz Doppler Width (D1 at 300 K) 361 MHz Melting Point 28.4 0C Vapor Density at 1500C 2.1841×1014 atoms cm−3

For liquid phase, 4006.048 log P = 8.221 27 − − 0.000 601 94 T − 0.196 23 log T (2.2) 10 v T 10 where Pv is vapor pressure in Torr and T is absolute temperature in K. But this model is a rough guide rather than a source of precise vapor-pressure values. Once the value of Pv and T are known we can calculate the vapor density by using gas model.

V apor density = Pv/kT where k is Boltzman’s constant and Pv is pressure in Pascal. The Figure 2.2 shows the vapor density and vapor pressure at various temperatures.

5 Figure 2.2: Vapor pressure and number density of cesium atoms.

6 Chapter 3

Theory of Polarization Spectra

3.1 Excitation Scheme

When an electron at the ground state absorbs energy equal to the difference in energy between ground state and excited state, the electron jumps to upper excited state which ultimately decays to the ground state by releasing the previously absorbed energy in the from of radiations of frequency ν (Fig.3.1).

Figure 3.1: Excitation of atoms.

2 In our experiment, cesium atoms are excited from the ground 6s S1/2 state to 2 2 the intermediate 6p P3/2 state with one photon and then ultimately to the 10s S1/2 2 level with another photon. The ground state of cesium is 6s S1/2 and has zero energy (Fig.3.2). We use two-photon two-color nano-second pulsed lasers to excite cesium atoms from the ground state to the final state. The first laser (pump laser) has wavelength of 852.112 nm which matches the wavelength for the transition 2 2 from the 6s S1/2 state to the intermediate 6p P1/2 state. The intermediate state has 30.4 ns lifetime. The wavelength of the second laser matches with the tran- sition from the intermediate state to the final state. Finally, these two photons 2 populate the final 10s S1/2 state. However, the electrons can not decay to the 2 2 6s S1/2 state directly from the 10s S1/2 state since S-S transition is forbidden. 2 2 Hence, the cascade fluorescence is observed from the 9p P1/2 to the 6s S1/2 state as shown in Figure 3.2.

7 Figure 3.2: Grotrian diagram showing the excitation and emission scheme.

3.2 Selection Rules

When a photon, whose energy is equal to the energy difference between two atomic levels, interacts with an ensemble of the atoms then the electrons of the atoms are excited to an upper level. Depending on the angular momentum J~ of the excited level it has magnetic sublevels ± mJ . The polarization state of the light determines which magnetic sublevels will be populated. If the interacting light is linearly polarized then population takes place for the magnetic sublevels for which change in magnetic quantum number is zero according to selection rule (∆m = 0). If the light is right circularly polarized it populates the magnetic sublevels with the selection rule ∆m = +1 and ∆m = -1 for the left circularly polarized light. Thus, it is a rule which summarizes the changes that must take place in quan- tum number of a quantum mechanical system to transit electrons between two states with certain probability. It is well known fact that transitions are not pos- sible between all possible pairs of energy levels. This means some transitions are forbidden while others are allowed by a set of selection rules. For electric dipole transition the magnetic quantum number changes by zero or unity. Selection rules specify the possible transitions among quantum levels due to absorption or emission of the electromagnetic radiation. In our experiment laser light interacts with the cesium vapor. The ground 2 1 state of cesium is 6s S1/2 and has two magnetic sublevels (± 2 ). The intermediate 2 1 3 2 6p P3/2 state has four magnetic sublevels (± 2 , ± 2 ) and the final 10s S1/2 state 1 has two magnetic sublevels (± 2 ). Figure 3.3 shows the possible transitions in

8 our experiment. Suppose we have right circularly polarized pump laser which

Figure 3.3: Selection rule.

2 2 excites electrons from the 6s S1/2 to the 6p P3/2 state. In the Figure 3.3, LCP refers to left circularly polarized light (σ−) and RCP refers to right circularly polarized light (σ+) respectively. If the probe laser, which excites the electrons 2 2 from the 6p P3/2 to the 10s S1/2 level, is right circularly polarized then atoms are not excited to the final state since it violates selection rule. However, with the presence of argon atoms these excited Cs atoms collide with the ground state argon atoms and this randomizes the populations in all four magnetic sublevels in the excited state. When this happens right circularly probe laser can excite atoms 2 to the 10s S1/2 state. This phenomena is explained in section 8.2 of chapter 8.

3.3 Alignment and Orientation

Figures 3.4 and 3.5 show the population distribution and physical illustrations for the alignment and orientation in the excited level. Each vector in the figures represents number of particles having same J~. If the charge distribution in atomic levels is such that the net angular momentum zero then it is called alignment. If the net angular momentum is not zero then it is called orientation. Thus, alignment and orientation measure the net angular momentum of electrons in the atomic level. The electric quadrupole component of the density in the excited level is called alignment while the magnetic dipole component of the density is called the orientation. Atoms show anisotropy when they are excited to upper level with a polarized light source. Fano and Macek [15] introduced the concept of alignment and orientation to observe the variation of the anisotropy of an atom during the light emission and introduced a general expression of the intensity of the polarized light

9 Figure 3.4: Aligned axially symmetric system. emitted in the right-angle geometry in terms of alignment and orientation. We can experimentally measure the amount of alignment and orientation in the excited state through the polarization spectra. The common geometry of an experiment of light-matter interaction is shown in Figure 3.6. Here xyz is collision frame and x0 y0 z0 is detection frame [16]. Collision frame has symmetry of excitation process of interest and detection frame, if chosen z-axis (quantization axis), will have cylindrical symmetry about the axis. Here geometry-relating angles are θ and φ but χ is the orientation of polarizer in front of the detector. Since z0 is direction of the observation vector, polarization vector lies on x0 y0 plane. Polarization vector can be defined as

~² = ˆi cosβ + ˆj i sinβ (3.1)

Here β defines the polarization of light to be detected. Hence β = 0 or π/2 for linearly polarized light and β = ± π/4 for circularly polarized light. The alignment can be written in terms of total angular quantum number J 0 and magnetic quantum number m0 of the final state as [15]

h3J 0 2 − J 0 2i X |a(m0 )|2[3m0 2 − J 0 (J 0 + 1)] hA i = z = , 0 J 0 (J 0 + 1) J 0 (J 0 + 1) m0 and the orientation can be written as 0 X 0 2 0 hJzi |a(m )| [m ] hO0i = q = q . 0 0 0 0 J (J + 1) m0 J (J + 1)

10 Figure 3.5: Oriented axially symmetric system.

3 The calculated values for alignment and orientation for J = 2 are, 5 hO i = √ o (2 15) 2 hAoi = 5

3.4 Density Matrix Formulation

The transition of electrons from lower state to higher state can be represented by density matrix element. For an ensemble of N atoms each with wave function | ψii, ensemble average of any observable A is given by,

1 XN X hAi = hψi | A | ψii = T r [ρA] = hn | ρA | ni. (3.2) N i=1 where ρ is the density matrix defined by,

1 XN ρ = | ψiihψi | . (3.3) N i=1 The density matrix contains the distribution of atoms among the various sublevels, since the probability of finding an atom in any state | ni is simply hn | ρ | ni. We will use density matrix formulation to calculate the polarization degree of 2 2 6s S1/2 → 10s S1/2 transition in Cs. When electrons jump from initial state with quantum numbers |  mi to the final state with quantum numbers | j0 m0 i, the corresponding density matrix element is given by Wigner-Eckart theorem

11 Figure 3.6: Collision and detection frame.

0 0 k 0 0 0 k hj m | Tq | j mi = C(j k j ; m q m ) hj | T | ji (3.4) where hj0 | T k | ji is the reduced matrix elements which are independent of m and m0 . C (...) is the Clebsch-Gordan coefficient and it contains the conservation of angular momentum, therefore vanishes unless m0 = q+m. For the electric dipole transition, the electric potential operator Vˆ is

Vˆ = eE.~r~ = eE ².~rˆ (3.5) where² ˆ is the polarization direction of electric field and r is the distance of electron from the nucleus. Then, the density matrix element is

hf | ².~rˆ | ii. (3.6)

For the given transition A in Figure 3.7 , the density matrix can be written as

3 1 1 1 h − | Vˆ | − i (3.7) 2 2 2 2

12 Figure 3.7: Transition of electrons from s to p level.

ˆ In spherical polar coordinate, V = rq where,

 √  −1/ 2 (x + iy) if q = 1; r = z if q = 0; (3.8) q  √  1/ 2 (x − iy) if q = -1.

For left circularly polarized light q = -1 and for right circularly polarized light q = +1 and for linear polarized light along z-axis q = 0.

3.5 Circular Polarization Spectra

3.5.1 Linear and Circular Polarization of Light Polarization is the state of vibration of electric field in the electromagnetic ra- diation. If the vibration of electric field is confined in a single plane though its magnitude and sign vary with time then such radiation is called plane or linearly polarized [19]. The plane contains both electric field vector E~ and propagation vector ~k. If the light is composed of two plane waves of equal amplitude by dif- fering in phase by 900, then the light is said to be circularly polarized. Circularly polarized light consists of two perpendicular electromagnetic waves with equal amplitudes but 900 out of phase. The tip of the electric field vector appears to be moving in a circle as it approaches the observer. If the electric vector of light coming towards you appears to rotate in clockwise direction then light is said to be right circularly polarized. If counterclockwise then it is left circularly polar- ized (Fig. 3.9). Suppose we have two linearly polarized light waves with same frequency and are moving in the same direction so that their electric field vectors are collinear. They superpose to give a resultant linearly polarized light. Let

13 Figure 3.8: Combination of two linearly polarized waves to form a resultant lin- early polarized wave.

~ ~ Ex(z,t) and Ey(z,t) are electric field vectors in the x and y directions of two waves moving in the z-direction. Then the orthogonal optical disturbances can be represented as

~ Ex(z, t) = ˆi E0x cos(kz − ωt) (3.9) ~ Ey(z, t) = ˆj E0y cos(kz − ωt + ²) (3.10) where ² is the relative phase difference between the waves. The resultant electric vector is then given by

~ ~ ~ E(z, t) = Ex(z, t) + Ey(z, t) (3.11)

When ² = 0 or + 2nπ , the waves are in phase

~ E(z, t) = (ˆi E0x + ˆj E0y) cos(kz − ωt) (3.12) where (ˆi E0x + ˆj E0y) is the amplitude. Since the amplitude is constant, it is still linear. Hence the two waves superpose to give another linear polarized wave (Fig 3.8). Consider another case where amplitudes of two waves are same i.e. E0x = E0y = E0 and phase difference(²) is - π/2 + 2mπ , where m = 0, ± 1, ± 2

14 Figure 3.9: Combination of two linearly polarized waves to form circularly polar- ized wave.

Figure 3.10: Circular polarization of light.

~ Ex(z, t) = ˆi E0 cos(kz − ωt) (3.13) ~ Ey(z, t) = ˆj E0 sin(kz − ωt) (3.14) The resultant E~ field is,

~ E(z, t) = E0[ˆi cos(kz − ωt) + ˆj sin(kz − ωt)] (3.15) q ~ ~ The magnitude of the amplitude is E = (E.E) = E0 constant but the direction of E~ is time varying, and it is not restricted to a single plane. At t = 0

15 ~ ~ Ex = ˆiE0 cos(kz) Ey = ˆjE0 sin(kz) ~ ~ At later time t = kz/ω, Ex = ˆiE0 and Ey = 0, so the electric field is along x-axis. The tip of the electric vector is moving clockwise as seen by the observer towards when the wave is moving. This is right circular polarized light (Fig 3.10). If ² = π/2 + 2mπ , m = 0, ± 1, ± 2 then,

~ E(z, t) = E0[ˆi cos(kz − ωt) − ˆj sin(kz − ωt)] (3.16)

In this case the E field rotates counterclockwise, so it is left circularly polarized light.

3.5.2 Intensity and Circular Polarization Degree When two circularly polarized light source pass through cesium vapor, Cs atoms 2 are excited into the final 10s S1/2 state according to selection rule given in section 2 3.2 and so the fluorescence can be detected from the 9p P1/2 state. The circular 2 2 2 polarization degree for the 6s S1/2 → 6p P3/2 → 10s S1/2 transition can be ex- pressed in terms of alignment and orientation. The intensity of fluorescence can be written as [16], n 1 1 0 I(θ, χ, β) = I 1 − h(2)(J, J )P (cos θ)hA i 3 0 2 2 0 3 0 + h(2)(J, J )hA i sin2 θ cos 2χ cos 2β 4 0 o 3 0 + h(1)(J, J )hO i cos θ sin 2β (3.17) 2 0 ³ ´ 3 2 1 nd (1) 0 where P2(cos θ) = 2 cos θ − 2 is the 2 rank Legendre polynomial. h (J, J ) and h(2)(J, J 0 ) are geometric factors which depend on the initial and final angu- lar momenta. hA0i and hO0i are the expectation values of the alignment and orientation, respectively. Since the detection is at right angle to the propagation vector of laser light, χ = π/2 n 1 1 0 I(θ, β) = I 1 − h(2)(J, J )P (cos θ)hA i 3 0 2 2 0 o 3 0 + h(1)(J, J )hO i cos θ sin 2β (3.18) 2 0

16 For circular polarization, θ = 00 n 1 1 0 I(β) = I 1 − h(2)(J, J )hA i 3 0 2 0 o 3 0 + h(1)(J, J )hO i sin 2β (3.19) 2 0 For circular polarization with positive helicity, β = π/4, n + 1 1 0 Iσ = I 1 − h(2)(J, J )hA i 3 0 2 0 o 3 0 + h(1)(J, J )hO i (3.20) 2 0 For circular polarization with negative helicity, β = −π/4 then, n − 1 1 0 Iσ = I 1 − h(2)(J, J )hA i 3 0 2 0 o 3 0 − h(1)(J, J )hO i (3.21) 2 0 The total signal we detect when both lasers have positive helicity is the combina- σ+ σ+ tion of I1 I2 If (Fig. 3.11). Thus,

σ+ σ+ σ+ S = I1 I2 If

σ− σ+ σ− S = I1 I2 If

Figure 3.11: Energy level diagram of Cs.

Then polarization degree can be written as

17 Sσ+ − Sσ− P = . (3.22) C Sσ+ + Sσ− Thus we can write circular polarization degree in terms of alignment and orienta- tion as,

(1) 0 3h (J, J )hOoi PC = (2) 0 (3.23) 2 − h (J, J )hAoi where,

s 3 1 J 0 + 1 5 h(1)( , ) = q = 2 2 J 0 (J 0 + 1) 3

3 1 J 0 + 1 −5 h(2)( , ) = − = 2 2 2J 0 − 1 4

Substituting h(1) and h(2) into Equation 3.23 we can re-write circular polarization degree as √ 15hOoi PC = (3.24) 2 + 5/4hAoi 3.6 Fine and Hyperfine Interactions

3.6.1 Fine Structure Fine structure is a result of the coupling of the orbital angular momentum L~ and spin angular momentum S~. There is an additional energy due to this coupling in the atomic level. The coupling of L~ and S~ combine to give a total angular momentum J~ as

J~ = L~ + S~ ~ whereq J takes the values | L − S | < | J | < | L + S | and its magnitude is j(j + 1). The only valence electron of cesium moves around the orbit and cre- ates internal magnetic field which interacts with electron’s magnetic field and orients the spin. This interaction is called the spin-orbit interaction where the spin angular momentum S~ and the orbital angular momentum L~ combine and

18 Figure 3.12: Hydrogen atom from the electron’s perspective. create a total angular momentum J~. The fine splitting is caused by an additional magnetic field which results from the interaction between the magnetic moment and the electronic field. To understand how the fine structure arises in atoms, let us describe the fine structure splitting in hydrogen atom since Cs is a hydrogen like atom. If we consider from the electron’s point of view, nucleus of charge + e moves around the negative electron with period T (Fig: 3.12). This nucleus sets up a current I = e/T and produces magnetic field B~ at the location of electron at a distance r. This magnetic field exerts torque on the spinning electron and aligns its magnetic moment ~µ along the direction of the field. The Hamiltonian of the system can be written as [20],

H = −~µ.B~ (3.25) where B~ is the magnetic field vector and the magnitude is

µ I B = 0 (3.26) 2r ~ 2 Using L = m~v.~r , I = e/T and µ0 = 1/²0 c , we can re-write Eq 3.26 as

~ 1 e ~ B = 2 3 L (3.27) 4π²0 mc r e ~ But the electron’s magnetic moment is ~µe = − m S. Substituting the values of µe and B~ in Equation 3.25 we can re-write Hamiltonian as

2 e 1 ~ ~ H = 2 3 S.L (3.28) 4π²0 mc r

19 Since the electron is not in the inertial frame, Thomas precession is used to correct this calculation and Hamiltonian of spin orbit interaction then is

2 e e ~ ~ Hso = 2 3 S.L (3.29) 8π²0 mc r In the presence of spin orbit coupling, Hamiltonian does not commute with L~ and S~ but commute with L2 , S2 and J. Thus,

1 L.~ S~ = (J 2 − L2 − S2) (3.30) 2 ¯h2 and its eigen value is 2 [j(j + 1) − l(l + 1) − s(s + 1)]. In this case s = 1/2 and

1 1 h i = (3.31) r3 l (l + 1/2) (l + 1) n3a3 Thus, the energy correction is

2 0 0 E n [j(j + 1) − l(l + 1) − s(s + 1) − 3/4] E = hH i = n (3.32) so so mc2 l (l + 1/2) (l + 1) The relativistic correction of energy is

2 0 E 4n E = − n [ − 3] (3.33) r 2mc2 l + 1/2

0 0 Combining Er and Eso gives us complete fine structure formula as

2 0 E 4n E = n (3 − ) (3.34) fs 2mc2 j + 1/2 When this is combined with the Bohr’s formula, we get energy level of hydrogen including the fine structure splitting as

13.6 ev α2 n 3 E = − [1 + ( − ] (3.35) nj n2 n2 j + 1/2 4 where α is the fine structure constant. Its numerical value is α = e2/¯hc = 1/137.037 which determines the magnitude in fine splitting. The fine structure breaks the degeneracy in l. Thus, different allowed values of l do not carry same energy.

20 3.6.2 Hyperfine Structure In the case of atoms with non-zero nuclear spin, the fine structures further split. The motion of an electron produces an internal magnetic field at the position of nucleus. This field interacts with the magnetic moment of nucleus and gives rise to further splitting in the fine structure. This is called hyperfine splitting. The interaction between magnetic moment of the nucleus and the magnetic fields of the electron shells orients the nuclear spin of atom. When the total angular mo- mentum J~ couples with nuclear spin I~ then a space-fixed total angular momentum F~ is formed where,

F~ = I~ + J~ (3.36) and F takes the values from J+I to ..... J-I. The hyperfine interaction energy WF

Figure 3.13: Hyperfine splitting of the levels of 133Cs.

21 is given by [4]

1 3 K(K + 1) − 2I(I + 1)J(J + 1) W = hAK + hB 2 (3.37) F 2 2I(2I − 1)2J(2J − 1) where K = F (F + 1) − I(I + 1) − J(J + 1) and A and B are the dipolar and quadrupolar coupling constants respectively. The first term of this equation arises from the interaction between the nuclear magnetic moment and electronic mag- netic dipole moments. The second term arises from the interaction between the electronic charge distribution and the electric quadrupole moment. The hyperfine 2 2 2 splitting of 6s S1/2, 6p P3/2 and 10s S1/2 states of Cs are shown in Figure 3.13. 7 For cesium the nuclear spin I = 2 and the coupling constants [21] for each energy state is given in the Table 3.1.

Table 3.1: Table for coupling constants used in the calculations.

Level A (MHz) B (MHz) 2 6s S1/2 2298.157 0 2 6p P3/2 50.34(6) −0.38(18) 2 10s S1/2 63.2 0

3.7 Hyperfine Depolarization Effects on Circu- lar Polarization Spectra

Hyperfine splitting in the atomic levels decreases the polarization degree since the excited electrons from these level decay to the ground level by giving radiation of different wavelengths to interfere each other. Since there are electrons in hyperfine levels the total intensity is the summation over the square of amplitudes of all possible transitions. Time dependent Schrodinger equation can be used to solve this type of non-stationary states. However, Fano and Macek has shown a simple formula for the splitting caused by the hyperfine structure [15-18]. The formula includes an oscillating time dependence in the alignment and orientation and can be written as

(2) hAo(t)i = hAoig (t), (3.38) and

(1) hOo(t)i = hOoig (t) (3.39)

22 where g(1)(t) and g(2)(t) are orientation and alignment hyperfine depolarization coefficients respectively. It is given by [22],

X 0 (k) (2F + 1)(2F + 1) 2 0 g (t) = W (JFFJF ; Ik) cos (ωF 0F t) (3.40) F,F 0 (2I + 1) where ωF 0F is the frequency splitting between two hyperfine levels and W(...) is Racah coefficient. The depolarization coefficient depends on the precession time

Figure 3.14: Overlap time of the pump and probe lasers. of J~ about F~ and affects the excited state. If the atoms are excited to the final state before this precession time takes place then depolarization effect due to unobserved hyperfine structure can be minimized. In that situation, for example, at t = 0 g(2) = 1 but after the hpf precession time takes place g(2)6= 1. This is time dependent alignment. The degree of circular polarization with hyperfine depolarization coefficient can be written as [16], √ (1) 15hO0ig (t) PC = (2) (3.41) 2 + 5/4hA0ig (t)

The g(1)(t) and g(2)(t) coefficients decreases from their maximum value of 1 at t = 0 and start to oscillate between 1 and -1 as the t is greater than 0. This oscillation introduces time evolution in all state multipoles. The temporal overlap time T (Fig. 3.14) of the 5-6 ns can be made smaller than hyperfine precession time 2 2 in the excited 6p P3/2 state. The hyperfine precession time in the excited 6p P3/2 state Cs is 1.6 ns. The overlap time of our pulsed laser is about 1.1 ns. Thus, due to comparable time scales and unknown laser pulse shape, all states multipoles are expected to be perturbed by hyperfine interaction and which in turn decreases the circular polarization degree. Using the pump-probe technique we can reduce the hyperfine depolarization effect. In this technique, we can control the overlapping time of two lasers. If the overlap time of the pulses is shorter than the inverse of the greatest hyperfine frequency ( ωFF 0 ) component in the intermediate state

23 then it is possible to excite the atoms to final state before the I and J coupling creates the hyperfine structure in the intermediate state. This would freeze the electronic alignment and gives us degree of polarization without any hyperfine depolarization effect.

24 Chapter 4

Experimental Apparatus

This section gives a detailed description of the experimental set-up and optical system for the circular polarization spectroscopy measurement. We will discuss about Lasers, circular polarization set-up, cesium cell in three sub-divisions.

4.1 Lasers

Two types of pulsed lasers are used in this research and they are a) Nd:YAG laser and b) Dye laser

4.1.1 Nd:YAG Laser We use Continuum Surelite Nd:YAG pulse laser for pumping our home-made dye lasers. The Nd:YAG ( neodymium doped yttrium aluminum garnet) operates at the second harmonic generator to produce wavelength at 532 nm with power of 4.5 W. The pulse width of the laser is about 4-6 ns with repetition rate of 20 Hz. We use beam splitters (CVI W1-PW1-1012-C-532-45UNP) to split the power of YAG laser to few percent and direct it to the dye laser apparatus. These beam splitters have antireflective (AR) coating on one side to prevent ghost reflection entering the laser cavity. Each of the beam splitters has 9% power transmission coefficient. The power of YAG laser at the exit port is 4.5 W but it is 0.2 W after the first and 0.18 W after the second beam splitter. A power meter (Molectron Detector Inc, PowerMax PM10V1) is used to measure the power of the YAG laser. It has maximum pulse energy density resistance of 5 J/cm2 (1 GW/cm2)

4.1.2 Dye Laser Using the YAG laser we produce pump dye laser of wavelength at 852.114 nm and probe dye laser of wavelength 603.412 nm. We use organic dye to produce dye laser. Dye cuvette in the laser cavity contain organic dye diluted in methanol

25 Figure 4.1: Experimental apparatus. with appropriate concentration. Our dye lasers are home-built and are constructed in Littman-Metcalf cavity configuration designed at grazing incidence to achieve a wide tunability and narrow bandwidth operation [27]. These dye lasers are pumped by Nd:YAG (Fig. 4.2). The geometry of grazing incidence dye laser cavity is shown in Figure 4.3. Each dye laser cavity has a grating, a dye cell, a cylindrical lens, feedback mirror, and an output coupler. The dye cuvette is tilted to avoid reflection from the back surface of the cell.

Dye Laser 1

2 A laser of wavelength 852.112 nm excites cesium atoms from the 6s S1/2 state 2 to the 6p P3/2 state. To produce the laser of the desired wavelength, we use an organic dye LDS 867 (Exciton Inc) diluted with methanol and concentration of

26 Figure 4.2: Pumping scheme of experiment. the oscillator is 84 mg/L. The wavelength of laser 1 (pump laser) is 852.112 nm with tunability range of 830-920 nm and the average output power is 1.3 mW. Since the dye degrades very fast we use a dye circulator (Spectra Physics Model 376) to stabilize the power of the pump laser. We use an AR coated flow cell (NSG Precision Cells Inc. T-524) with 2 ml capacity. It is made of quartz and light path length is 8 mm. A gold-coated grating (Edmund Industrial Optics, Y55-261) is used in the cavity of laser 1. Its dimension is 30 mm x 30 mm with 1200 groves/mm and has an efficiency of 75% at a blaze wavelength of 750 nm. Output coupler is a wedged window (CVI LW-2-1037-C) which prevents interference between the reflections from the front and back surfaces. There is a cylindrical lens with a focal length of 5.08 cm to focus the Nd:YAG laser at the edge of dye cell. To avoid heating of the dye we lightly focussed the YAG on the cell. A near infrared coated mirror (Thorlabs Inc; BB1-EO3) is used as a tuning mirror in the cavity. Its reflectivity is 99.9% in the range of 750-1150 nm. It is mounted on a precision mount, which gives flexibility to tune the laser to a desired wavelength.

27 Figure 4.3: Littman-Metcalf cavity configuration.

Dye Laser 2

133 2 2 To excite Cs from the 6p P3/2 state to the 10s S1/2 state we need a laser of wavelength of 603.409 nm. We use Rhodamine 640 Perchlorate (Exciton Inc.) diluted with methanol to produce dye laser of the desired wavelength. We have an oscillator and an amplifier for the dye laser 2. The oscillator uses the dye with concentration of 141.8 mg/L and the amplifier uses the same dye with con- centration of 18.9 mg/L. The amplifier approximately amplifies the laser more than 10 times. The dye concentrations of laser are selected from Exciton catalog (www.Exciton.com). The wavelength tunability for the laser 2 is in the range of 580-650 nm. The average power of the laser is 0.2 mW before the amplifier and 6 mW after the amplifier. The powers of the dye laser are measured with a power meter (Molectron Detector Inc., PowerMax PS19), which has a maximum pulse density resistance of 50 mJ/cm2 (1 MW/cm2) We use quartz AR coated dye cells (T-509) for the oscillator and amplifier. The capacity of dye cell is 3.2 ml with a light path of 8 mm. A holographic grating (Edmund Industrial Optics, Y43-215) is used in the cavity of laser 2. The dimension of grating is 12.5 mm x 25 mm with 1200 grooves/mm. The efficiency of this grating is 55%. We use cylindrical lens and output coupler similar to the one used in dye laser 1 cavity. The laser cavity has a broadband mirror (Thorlabs Inc. BB1-EO2) with 99.7% reflectivity in the range of 400-900 nm. This mirror is mounted on an ultrastable kinematic mirror mount (Newport 610 series) and has a coarse and a fine adjustment knob for vertical and horizontal axes. The horizontal fine scale is replaced with a stepper motor driver (Ardel Kinematics Corp, Motor Mike) to remotely tune the laser wavelength. Each motor driver step corresponds to a 0.01 nm change in the wavelength. Wavelength of the laser is measured by a wave meter (Coherent WaveMaster), which is sensitive to a wavelength range of 380 nm to 1095 nm with an accuracy of 0.001 nm. It has a

28 power resistance of 100 mW. Infrared viewer (Electrophysics, Electroviewer 7215) or infrared card (Throlabs Inc., IRC3) is used in order to align the laser 1. We used irises after the output coupler of pump laser and probe laser amplifier so that we can eliminate unwanted light and pick up the central portion of the laser beams. In each of the laser path we use two lenses to produce fine collimated beam. These two converging lenses are separated at a distance equal to the sum of their focal lengths. The height of each laser is the same so that they can overlap within the interaction region of the cesium cell. All mirrors used in the laser path are dielectric mirrors (Thorlabs Inc; B1-EO3 and BB1- EO2) chosen for their specific reflectivity wavelength range.

4.2 Free Spectral Range

The free spectral range of an optical resonator cavity is the frequency spacing of its axial resonator mode. It basically determines the minimum spacing of the lines in the optical spectrum of the laser output. The free spectral range of laser cavity can be calculated by the following equation, c FSR = 2nd where c is the speed of light, d is the cavity length, and n is the the refractive index of the medium. Our laser cavities are open to air so the refractive index for air is 1. Thus, c FSR = 2d The cavity length of laser 1 is 12 cm and 11.5 cm for laser 2. Thus the calcu- lated FSR of dye laser 1 and dye laser 2 cavities were 1.25 GHz and 1.30 GHz respectively.

4.3 Circular Polarization Set-up

The Nd:YAG laser is more than 95% vertically polarized along laboratory z-axis. Since the dye laser cavities do not change the state of the polarization of the YAG laser the output beam of the the dye lasers are vertically polarized. However, the output beam may not perfectly vertically polarized because of the presence of background fluorescence in it. To produce a higher degree of vertical polarization we use a Glan Thompson polarizer (Thorlabs Inc; GTH10M) in the path of both lasers. Our Glan Thompson Polarizer has contrast ratio of 1:100,000 so it helps to produce a higher degree of vertically polarized laser light. Both dye lasers are collinear and are carefully aligned so that they perfectly overlap at the center of the oven, which has a cesium cell at the center. Using infrared card when blocking and unblocking the laser 1 beam checks overlapping of two lasers. This is done

29 before they enter the oven from both sides of the oven windows. We use liquid crystal variable retarder (LCVR) in the path of laser 2 to change the polarization state of the laser. We use a quarter wave plate (Thorlabs QWP 600) in the path of laser 2 and another quarter wave plate (Thorlabs (QWP 890) in the path of laser 1 to obtain circularly polarized light. When the fast axis of the quarter wave plate is at +450 with the vertical then the emergent beam has positive helicity but if the quarter wave plate is at - 450 with the vertical the emergent beam has negative helicity. LCVR changes the helicity of light. When both of the lasers have positive helicity then the recorded signal intensity is termed as Sσ+ but if the lasers have opposite helicity then the recorded signal intensity is termed as Sσ− . Nematic type liquid crystal molecules fill the LCVR cavity. Nematics molecules have a common axis of alignment and weak external agents can rotate their opti- cal axis. Electrical components are attached with the cavity and it is connected to an interface to remotely control the alignment of the nematics. By applying a certain voltage to LCVR we can change the slow axis and fast axis to achieve a sought polarization. We apply nearly 2 V to obtain signal Sσ− and about 6 volt to obtain the signal Sσ+ . LCVR is mounted on a high precession rotation mount (Thorlabs Inc, PRM1) to attain a high-level polarization. The contrast ratio (CR) of LCVR is very important in the polarization mea- surement. To measure CR we pass laser 2 in succession through the first GTP, oven, and then through the second GTP on to a photodiode (Thorlabs Inc., SM05PD1A). The photodiode is connected to oscilloscope (HP 5422A). At first laser 1 is blocked and laser 2 is attenuated using neutral density filters (Thorlabs Inc., FW1) so that photodiode is not saturated. Once the photo diode shows lin- ear response to the incident light the voltage on the LCVR is gradually changed to obtain the minimum signal observed on real time using oscilloscope. The LCVR is remotely controlled by a computer driver software program (Meadowlark Optics, Cell Drive 3000), which changes the voltage to obtain the sought polarization. Since the photodiode is placed right after the second GTP, we can obtain the optimum voltage for the LCVR by observing the intensity of dye laser 2 on the oscilloscope. We gradually change voltage to get minimum signal on the oscillo- scope screen. The fine adjustments are done by slightly rotating the LCVR with its high precision mount to obtain higher degree of polarization. Then the ratio of observed maximum signal to minimum signal gives the contrast ratio of LCVR. The contrast ratio of LCVR was measured to be 1:4000 but it slightly fluctuates according to temperature of LCVR and position of GTP. Contrast ratio is checked every time before taking any polarization data. The optimum voltage value for LCVR is then used in a LabVIEW program (National Instruments, LabVIEW 7.0 Software) and its Data Acquisition Board (DAQ) (National Instrumental, PCI 1200) to obtain the polarization spectrum.

30 4.4 Cesium Cell

The interaction region in our experiment is cesium cell filled with argon buffer gas at various pressures. These cesium cells are cylindrical in shape and are made of pyrex glass. The length of our cell is 5.08 cm with diameter 2.54 cm. The cell is evacuated to a low pressure (approx.10−7 Torr). It is then baked under vacuum at high temperatures for 1-2 days to remove absorbed gases and surface contaminations. After baking, a very small amount of cesium (about a few tens of milligram) is introduced into each cell using a torch and then argon gas is added at pressures of 5, 30, 60 and 100 Torr. Pressure gauge is used to measure the pressure inside the cell. After this, each cell is sealed off and separated from the vacuum system, leaving behind a small stem on the cylindrical surface. The stem is made as small as possible to fit inside the oven. There is a possible change in buffer gas pressure while sealing the cell off from the vacuum. Thus, the glass at the pinch off must be locally heated to a temperature of over 500 0C. This is completed within few seconds but it is inevitable that some the gas in the cell is heated during the process. Some of the heated gas escapes to maintain the constant pressure within the cell at that temperature. But when the temperature falls to room temperature the pressure would fall somewhat below the intended pressure. In contrast, the stem-shrinking process tends to increase the pressure inside the cell as volume will be less. Since the diameter of stem is very small compared to the volume of the cell it will have no significant effect on the pressure measurement. The cell is now placed

Figure 4.4: Cesium cell. inside a hollow cylindrical aluminum oven. The oven has two windows at two ends to pass the laser light from two opposite sides. The diameter each of theses windows is 1.27 cm. There is the third window perpendicular to the first two to detect the cascade fluorescence signal. The photographs of the oven and cell are shown in Figures 4.4 and 4.5. The oven rests on two stands. A concave mirror is mounted inside oven on the opposite side of the observation window to

31 Figure 4.5: View of the oven wrapped with heating tape and aluminum foil. collect maximum signal at detection window. A K-type thermocouple (Omega, KMQSS-020U-60) is attached with the cell to monitor its temperature. The oven is uniformly wrapped with a flexible flat heavy insulated samox type electric heating tape (Cole Parmer, 36050-20), fiber flax , and aluminum foil around it to ensure uniform temperature inside the cell. The thermocouple and heating band are connected to a temperature controller (Cole Parmer, Digi-Sense EW-89000-00) to stabilize the temperature of the cell. The accuracy of the temperature controller is ± 0.10C. We use pure cesium cell as well as cells with argon buffer gas at pressures of 5 Torr, 30 Torr, 60 Torr, and 100 Torr. The atom density in the cell varies with temperatures (Fig. 2.2).

32 Chapter 5

Overview of Experimental Measurement

This chapter gives some details of experimental measurement overview. We used Nd:YAG laser of wavelength 530 nm to pump the home-built dye lasers. These dye lasers have wide tunability range. The wavelength of dye laser 1 was set at 852.112 nm and the wavelength of dye laser 2 at 603.409 nm. But in our experiment, we had difficulty measuring the wavelength of dye laser 1 because of its low power. Hence we observed Cs D2 signal by using a combination of interference filter and color glass filter and obtained peak signal on the oscilloscope. The maximum signal corresponds to the wavelength of 852.112 nm of laser 1. Also we calibrated the micrometer of the dye laser 1. The calibration graph is shown in Figure 5.1. Once

Figure 5.1: Micrometer tuning curve of laser 1. we found the resonance wavelength for Cs D2 line we changed the interference filter for two photon transition and observed two photon signal. The two-photon signal was weaker than one-photon signal as expected. To reduce the noise we covered

33 the signal cable with copper braided wire and PMT housing was well covered from outside by black clothes and black tape so that no scattered light could enter the PMT. Using GTP and quarter wave plate we produced circularly polarized light. Remotely controlled LCVR was used to change the helicity of the circularly polarized light. We applied 6 volts to produce circularly polarized light with the same helicity and about 2 volts to produce light with opposite helicity. By tuning the wavelengths of laser 1 and laser 2 we obtained double resonance condition. Gatewidth of the Boxcar averager/integrator was so adjusted that the signal was 2 2 averaged over the gate width. The fluorescence signal from 9p P1/2 state to 6s S1/2 state was detected and was measured using takedata3 sub.vi program when the ¯ light with same helicity light and with opposite helicity entered the Cs-cell. The signal was amplified 25 times by two stage amplifier. By using takedata3 sub.vi ¯

Figure 5.2: Geometry of the experiment program, we measured the signals. Before taking real polarization data we set the sensitivity of the boxcar averager/integrator at its maximum and then measured the baseline by blocking laser 1 using takedata3 sub.vi program. This baseline ¯ was provided to takedata3 sub.vi program to be subtracted from the real data. ¯ Before taking real data the power of both dye lasers were checked. If the power of dye lasers were low then dyes were refreshed. Also contrast ratio of LCVR was checked in daily basis before taking polarization data. Good contrast ratio ensures the good polarization data. Once the CR was good, we took polarization data. Each polarization data was the average of 1000 data points. When two lasers have the same helicity, the measured signal is Sσ+ and Sσ− when they have opposite helicity. We cal- culated polarization degree for each pair of Sσ+ and Sσ−. Final polarization was the average of several polarization degrees. We can use Wigner-Eckart theorem to 2 2 2 calculate circular polarization degree in the 6s S1/2 → 6p P3/2 → 10s S1/2 tran- sition. The Clebsch-Gordan coefficient for the related transition is given in Table

34 B.2 in Appendix B. We calculated 100% polarization in pure cesium without any systematic effect. However, it is 60% if we consider only hyperfine depolarization effect. To minimize the hyperfine depolarization effect one can use a pump-probe technique. Using this technique one can control over the overlapping time of pump and probe laser. If hyperfine setting occurs the overlap time of the pulses must be smaller than the inverse of highest hpf frequency splitting because atoms will be excited to final state before hyperfine occurs in intermediate state. The shortest precession time in cesium is about 1.6 ns. We shorten the overlap time to minimize the hyperfine effect.

Table 5.1: Some important parameters involved in the experiment.

PY AG 1.85 W

PL1 1.0 mW

PL2 (oscillator) 2 mW

PL2 (amplifier) 5.6 mW 2 1/2 6p P1/2 → 6p P3/2 852.112 nm 2 2 6p P3/2 → 10s S1/2 603.409 nm 2 2 9p P1/2 → 6s S1/2 361.730 nm L1 84 mg/L(Methanol) For 10S L2(oscillator) 114 mg/L(Methanol) L2(amplifier) 16.83 mg/L(Methanol) Gatewidth of the signal 50 ns

In the absence of Ar buffer gas the expected polarization degree of the two- 2 2 2 photon polarization process in the 6s S1/2 → 6p P3/2 → 10s S1/2 transition can be calculated from Equation 3.22 by using the Clebsch-Gordon coefficients listed in the Table B.2 in Appendix B.

35 Chapter 6

Signal Detection Scheme

2 Pump and probe lasers populate the Cs atoms in the 10s S1/2 state. These ex- 2 2 cited atoms can not decay from the 10s S1/2 state to the ground 6s S1/2 state. Instead, we detect the cascade fluorescent signal of wavelength of 361.730 nm from 2 2 the 9p P1/2 state to the ground 6s S1/2 state. For the detection of the cascade fluorescence we place a side-on photo multiplier tube (PMT) (Hamamatsu, R955) in front of the signal viewing port of the oven. When the fluorescent signal enters the PMT it hits the photo cathode to emit photoelectrons. These photoelectrons are multiplied by secondary emission and collected by anode as output signal. The spectral response of our PMT is 160-900 nm with quantum efficiency of 29% at 220 nm. Usually PMT is operated at -800 volts by using high power supply (Stanford research Inc, PS 350). A combination of interference filter (Coherent, 35-3045-000) and colored-glass filter (Coherent, UG11) is used to detect the fluo- 2 2 rescent signal from the 9p P1/2 state to the 6s S1/2 state. The interference filter passes 31% of the desired wavelength and has peak transmission wavelength at 366.7 nm at FWHM of 10.7 nm. We use colored glass filter to eliminate scattered YAG and dye lasers. The output of PMT was sent to two-stage amplifier (Stan- ford Research Inc. SR 240) to amplify the signal by a factor of 5 by each stage. We use 50 ohms coaxial cables between the stages and send the amplified signal to the boxcar averager/integrator (Standford Research Inc. SR 250). YAG pulses are picked up by a glass slide and are sent to a photodiode to trigger the boxcar. We monitored the output signal on oscilloscope while adjusting the gatewidth simultaneously. The gate width was 60 ns. The gatewidth is set in such a way that it detects signal within that width without any noise. We took last signal output of boxcar averager/integrator. This allows having a shot by shot analysis of signal. The signal output was sent through an outbreak box (National Instru- ments, CB-68LP) to Data Acquisition Board (DAQ) (National Instruments, PCI 6014), which converts analog to digital signal. The digital signal was processed by LabVIEW program (National Instruments, LabVIEW 7.0). The LabVIEW remotely controls the polarization and frequency scan of probe laser. Each step of stepper motor changes the wavelength by approximately 0.01 nm. The polar-

36 ization data taken at each frequency was the average of 1000 data points. We use takedata3 sub.vi program to take the circular polarization data at resonance ¯ frequency. To obtain resonance, we use micrometer subVI.vi program to change the wavelength of laser until we get maximum signal on the oscilloscope. Once the frequency of pump and probe lasers are at resonance, the polarization data are taken. The signal when two lasers are with same helicity and with opposite helicity are measured by takedata3 sub.vi program. Then the polarization degree is calculated ¯ by using Equation 3.22 in chapter 3.

37 Chapter 7

Systematic Effects

Systematic effects may change the polarization measurements. We tried our best to minimize the systematic effects. Major causes of systematic effect may be due to the temperature of the cell, power of lasers, and hyperfine splitting. The effect of hyperfine splitting on the polarization signal was discussed in section 3.7 of chapter 3. When our medium is dense and temperature is high then the radiation emitted by excited atoms are trapped inside the atoms and bounce back and forth between the absorbers. This is called radiation trapping and it decreases the degree of polarization. This is because the lifetime of excited atoms becomes longer and decay rate is slow. However, due to the short temporal overlap time (t =1.1 ns) of the pulses compared to the radiative life time of excited state (τ = 31.4 ns) approximately 3% of the excited atoms will have radiatively decayed during pulse. Therefore, the radiation trapping has negligible effect on the excited state. There is another type of systematic effect due to temperature and it is called Doppler broadening. This effect arises due to different speed of atoms at the given temperature. Since the atoms have different velocities they see different frequencies of incoming light. If the medium is excited by a laser of frequency, only a certain group of atoms can excite to higher state. The thermal 0 velocity of this group should satisfy the relation kv = ω -ω0 where ω0 is central frequency of Doppler line [23]. Those atoms at exact resonance with the laser excites to upper state and are ready to emit the light. But those atoms at off- resonance behave as normal absorber. This effect occurs when the field is weak and cause to narrow the dips in the Doppler broadened gauss line profile, which is known as ”hole burning” effect [24-26]. In our research we did not see any hole burning effect. In the absence of any depolarization effect, circular polarization is 100%. However, we expect 60% polarization when I and J coupling is considered with no other source of systematic effect. Hence, our circular polarization spectra 2 2 obtained for the 6s S1/2 → 10s S1/2 transition confirms that J’ = 3/2 level has no other systematic effect measurably alter the alignment and orientation level. Consequently both alignment and orientation hyperfine depolarization coefficient at t = 1.1 ns were taken into account in order to treat for the depolarization

38 influence of the unobserved hyperfine interaction. Besides these effects we may have birefringence of oven window and imperfection of polarizer which cause to change the direction, collinearity and uniformity of our laser. To minimize this effect we check this using the polarizer before and after the lasers entering the cell. The Glan-Thompson polarizer produces the pure linearly polarized light. We use two GTP to produce relatively parallel pump and probe laser. The extinction ratio of the polarizers is checked to purify the polarization of lasers. The lasers should align and overlap well within the cell. We can check the linearity, uniformity and alignment by sending the light with the use of a mirror to a few meters away from the oven. This is done in daily basis.

7.1 Effect of Power of Laser

To check power dependence polarization we used different density filters on the path of both of the lasers individually to measure the polarization degree. Our data was taken at low intensity regime at all times so that the intensity of the excitation field is much lower than the saturation limit (0.1 W cm−2). Our result showed that within statistical error there was no effect of power on the measured polarization. The power dependence polarization also checks the hole burning ef- fect. In such case the polarization degree is expected to be higher. The Figures 7.1 and 7.2 show the power dependence of measured polarization degree.

Figure 7.1: Effect of power of laser 1 on circular polarization degree.

39 Figure 7.2: Effect of power of laser 2 on circular polarization degree.

7.2 Effect of Temperature

We varied the temperature of cesium cell from 700C to 1500C to check any density 2 2 2 dependence on the polarization for the 6s S1/2 → 6p P3/2 → 10s S1/2 transition. The result within statistical error showed that there was no effect of temperature on the measured circular polarization. Our temperature dependence polarization measurement is shown in figure 7.3.

Figure 7.3: Effect of temperature of cell on the circular polarization degree.

40 Chapter 8

Rate Equation Analysis and Results

8.1 Rate Equation Analysis

We used rate equation analysis to analyze the circular polarization data and to extract disorientation cross-section. The population variations among the Zeeman 2 sublevels of the 6p P3/2 level due to subsequent collisions with buffer-gas can be expressed by a simple theoretical model using the rate equation analysis. These can be written as dN 3/2 = −(γ + Γ + Γ + Γ )N + Γ N + Γ N + Γ N + Γ , (8.1) dt 1 2 3 3/2 1 1/2 2 −1/2 3 −3/2 p dN1/2 = −(γ + Γ + Γ + Γ )N + Γ N + Γ N + Γ N + Γ 0 (8.2) dt 1 2 4 1/2 4 −1/2 2 −3/2 1 3/2 p , dN −1/2 = −(γ + Γ + Γ + Γ )N + Γ N + Γ N + Γ N , (8.3) dt 1 2 4 −1/2 4 1/2 2 3/2 1 −3/2 dN −3/2 = −(γ + Γ + Γ + Γ )N + Γ N + Γ N + Γ N , (8.4) dt 1 2 3 −3/2 1 −1/2 2 1/2 3 3/2 where γ, Γp, Γ1,2,3 are the radiative decay, pump pulse and collisionally induced transition rates. The net rate of change of the total population is dN(t) = −γ N + Γ + Γ 0 . (8.5) dt p p This process is illustrated in Fig 8.1. The time dependent total population density, alignment and orientation in the excited level can be written as

1 −γt N(t) = (Γ 0 + Γ )[ (1 − e )], (8.6) p p γ

4 1 −γat hA0(t)i = (Γp − Γp0 )[ (1 − e )], (8.7) 5 γa

41 Figure 8.1: Population mixing among the Zeeman coherence

1 1 −γot hO0(t)i = √ (3Γp + Γp0 )[ (1 − e )], (8.8) 15 γo where γa = γ + (Γ1 + Γ2 + Γ3) is the alignment and γo = γ + (Γ1 + Γ2 + Γ4) is the orientation decay rates due to the collisions. The population in each Zeeman magnetic sublevels can be written in terms of total population, orientation and alignment as √ 1 3 3 15 N3/2(t) = [N(t) + hA0(t)i + hO0(t)i], (8.9) 4 4 √5 1 3 15 N1/2(t) = [N(t) − hA0(t)i + hO0(t)i], (8.10) 4 4 √5 1 3 15 N−1/2(t) = [N(t) − hA0(t)i − hO0(t)i], (8.11) 4 4 √5 1 3 3 15 N (t) = [N(t) + hA (t)i − hO (t)i]. (8.12) −3/2 4 4 0 5 0 The alignment and orientation can be written in terms of the population in each magnetic sublevels of the J = 3/2 atoms as 4 hA i = [N − N − N + N ], 0 5 3/2 1/2 −1/2 −3/2

1 hO0i = √ [3N3/2 + N1/2 − N−1/2 − 3N−3/2], 15 and the total density N as

N = N3/2 + N1/2 + N−1/2 + N−3/2. The measured signals, integrated over the pulse width can be written as

Z T Z T σ+ 3 1 S = N−3/2dt + N−1/2dt, (8.13) 4 0 4 0

Z T Z T σ− 3 1 S = N3/2dt + N1/2dt (8.14) 4 0 4 0

42 The constants in front of the integrals are the Clebsch-Gordan coefficients. Sub- stituting Equations 8.9 to 8.12 into Equations 8.13 and 8.14 , we can re-write signal as + Γ + Γ 0 Γ − Γ 0 3Γ + Γ 0 Sσ = p p I + p p I − p p I , (8.15) 4 n 8 a 8 o

− Γ + Γ 0 Γ − Γ 0 3Γ + Γ 0 Sσ = p p I + p p I + p p I . (8.16) 4 n 8 a 8 o here,

Z T 1 − e−γ.t T 1 − e−γ.T In = .dt = + , 0 γ γ γ2

Z T 1 − e−γa.t T 1 − e−γa.T Ia = dt = + 2 , 0 γa γa γa

Z T 1 − e−γo.t T 1 − e−γo.T Io = dt = + 2 . 0 γo γo γo Then,

σ+ σ− 1 S − S = − (3Γ + Γ 0 ) I , 4 p p o

0 Γ + Γp Γ − Γ 0 Sσ+ + Sσ− = p I + p p I . 2 n 4 a Then, the circular polarization degree can be expressed in terms of the depo- larization cross section and the pressure of the buffer gas using kinetic energy relation

r1Z0 Pc = (8.17) 2 + r2Za where

0 3Γp + Γp r1 = 0 Γp + Γp and

43 0 Γp − Γp r2 = 0 Γp + Γp Note that 2 Γp = C (3/2, 1, 1/2; 3/2, 1, 1/2) = 3/4 and 2 Γp0 = C (1/2, 3/2, 1; -1/2, 1/2, 1) = 1/4 Thus,

r1 = 5/2 and r2 = 1/2

Substituting the values of r1 and r2 in Equation 8.17, we get

5Zo Pc = (8.18) 4 + Za where −γ t (1) n1 + (e 0 −1) o γg (t) T γo Zo(p) = (e−γt−1) γo 1 + T γ

−γat (2) n1 + (e −1) o γg (t) T γa Za(p) = (e−γt−1) γa 1 + T γ

Here, γa = γ + Γa and γo = γ + Γo are defined. The pressure dependencies in Γo (disorientation decay rate) and Γa (disalignment decay rate) are

Γo = ρAr ko

= ρAr σo v¯CsAr p = · σ v¯ . (8.19) kT o CsAr and

Γa = ρAr ka

= ρAr σa v¯CsAr p = · σ v¯ . (8.20) kT a CsAr where p is the buffer gas pressure, kT is the thermal energy constant, σo is the disorientation cross section. Γa can be written similarly. We used the fact that ko = hσovi, and hσovi may be factored so that ko = σohvi. We denoted hvi asv ¯CsAr which is the average velocities of the colliding Cs-Ar atoms over the Maxwell- Boltzmann distribution of relative velocities at the cell temperature.v ¯CsAr can be determined from the following Equation,

44 s 8kT v¯ = (8.21) CsAr πµ where T is absolute temperature of the cell, k is Boltzman’s constant, and µ is reduced mass of Cs-Ar atoms. Substituting Equation 8.19 in γo = γ+Γo and Equa- tion 8.20 in γa = γ + Γa, we can extract orientation and alignment depolarization cross-section by using weighted non-linear least square fitting as shown in Fig- ure 8.2. In the Equation 8.18, Pc is the measured polarization, p is pressure and σo is the only fitting parameter. We used previously measured value [5] of align- ment depolarization cross-section in our calculation. The numerical values used in the extraction of the σo and σa are given in Table 8.3. We use weighted nonlinear square fit because it minimizes the error to give best value of cross-sections. If the P 2 scatter is uniform, the least square regression minimizes (Ydata − Ycurve) and finds the best value of parameter. If the average amount of data is not uniform, a least square method tends to give undue weight to the points with large y-values and ignores points with low y-values. To prevent this we use following relative weighted scheme

X Y − Y ( data curve )2. Ydata 8.2 Polarization as a Function of Buffer Gas Pressure

2 The pressure dependence on our circular polarization spectrum for the 6s S1/2 → 2 2 6p P3/2 → 10s S1/2 transition is shown in Figure 8.2. Pressure dependence circu- lar polarization degree are listed in the Table 8.1. We used five different types of cesium cells filled with argon pressure ranging from 0 Torr to 100 Torr. Our data shows that there is strong dependence of argon pressure on the polarization de- gree when excited Cs-atoms collide with ground state argon atoms. Polarization degree is higher for lower argon pressure.

45 Figure 8.2: Weighted non-linear least square fit of the circular polarization spec- trum of Cs.

Table 8.1: Our work on the circular depolarization with argon buffer gas in the 2 2 6s S1/2 → 10s S1/2 transition.

Argon pressure (Torr) PC (%) [theory] PC (%) [experiment] error (%) 0 60.90 59.84 0.45 5 50.27 50.12 0.6 30 24.76 23.66 0.65 60 14.71 15.37 1 100 9.44 9.82 0.7

Table 8.2: The alignment and orientation depolarization cross-section of the 133 2 Cs 6p P3/2 J = 3/2 atoms.

2 2 σa (A˚ ) σo (A˚ ) σa/σo References 186(58) Bayram et.al [5] 151 (44) 1.23 (4) This work 238 192 1.24 Okunevich et.al [10] 288(72) 234(34) 1.23 (18) Guiry et.al [7]

46 Table 8.3: Some important parameters used in the experiment

τP3/2 30.473 ns −26 mAr 6.634×10 kg −25 mCs 2.206×10 kg −26 µAr−Cs 5.1 × 10 kg 7 −1 γ6P3/2 3.26×10 s vAr−Cs 48638 cm/s 1 Torr 133.32 Nm−2 −9 Tpulse (5-7) ×10 s −23 −1 kBoltzmann 1.3807×10 JK 0 −3 PV at 70 C 9.1791×10 P ascal R 8.3145 Jmol−1K−1 Melting pt of Cs 28.44 0C

47 Chapter 9

Conclusion and Future Directions

133 2 We experimentally studied the circular polarization spectrum of the Cs 6s S1/2 2 2 2 → 6p P3/2 → 10s S1/2 transition by using pump-probe technique. Since 10s S1/2 2 2 2 → 6s S1/2 transition is formidable, the fluorescence from the 9s S1/2 → 6s S1/2 transition were observed. We calculated and measured the state multipoles in the excited state of cesium atoms and measured the effect of nuclear hyperfine depo- larization as a function of overlap time of the pulses. The polarization degree in pure cesium is in excellent agreement with the theoretical and experimental values due to dependence of g(2) hyperfine depolarization coefficient. The polarization degree decreased as the pressure of buffer gas increased. This shows that there is strong depolarization effect due to collision of excited cesium and ground state argon. The systematic effects such as collisional broadening, radiation trapping were minimized by using pump-probe technique because we could control over the overlapping time of two lasers. From the circular polarization data we extracted 133 2 disorientation cross section in the Cs 6p P3/2 state. This is in good agreement with the theory. The future research with two-photon polarization spectroscopy will be with 133 2 different noble gases on collisional cross-section in the Cs 6p P3/2 state, the effect of magnetic field on polarization spectrum, and quantum beat spectroscopy to investigate the hyperfine levels in the excited cesium.

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50 Appendix A

Production and Detection of Linear and Circularly Polarized light Using Jone’s Matrices

As we know that when the electric field vector of electromagnetic wave vibrates in a single plane then it is called linearly polarized and if the tip of the electric field rotates in a circle then it is called circularly polarized. There are various ways to produce linearly polarized light.

1) When an unpolarized light passes through plane polarizer such as Glan Thomp- son polarizer then emergent beam is linearly polarized along the axis of the po- larizer.

2) When right circularly polarized light enters quarter wave plate, whose fast axis is at 450 with vertical, then the emergent beam will be linearly polarized along X-axis. We can" check# it using Jone’s matrix. The matrix for right circularly po- 1 larized light is and matrix for quarter wave plate whose fast axis at 450 −i " # 1 i with the vertical is √1 then, 2 i 1 " #" # " # 1 i 1 √ 1 √1 = 2 2 i 1 −i 0

" # 1 where is linearly polarized along x-axis. Similarly, when left circularly 0 polarized light passes through quarter wave plate, whose fast axis at 450 with vertical, the emergent beam is linearly polarized along y-axis.

51 " #" # " # 1 i 1 √ 0 √1 = 2 2 i 1 i i

" # 0 where is plane polarized along y-axis. When right circularly light passes i through quarter wave plate whose fast axis is vertical then it produces linearly polarized light at 450 with x-axis. " #" # " # 1 0 1 1 = 0 −i i 1 " # 1 where is linearly polarized light along 450 with x-axis. Similarly when 1 right circularly and left circularly polarized light combines they produce linearly polarized light. " # " # " # 1 1 1 + = 2 i −i 0

We have different ways to produce circularly polarized light as well.

1) When right circularly polarized light passes" through#" half wave# plate" it# produces 1 0 1 1 left circularly polarized and vice-versa. = which is 0 −1 −i i " # 1 0 right circularly polarized. Here is the matrix for half wave plate. 0 −1

2) When vertically polarized light passes through quarter wave plate, whose fast axis is at + 450 or - 450 with the vertical, then emergent beam will be right or left circularly polarized respectively. " #" # " # 1 i 1 1 √1 = √1 which is left circularly polarized. 2 i 1 0 2 i

" #" # " # 1 −i 1 1 √1 = √1 which is right circularly polarized. 2 −i 1 0 2 −i

To detect linearly polarized light we use Glan Thompson polarizer. The inten- sity of linearly polarized through GTP is maximum at two positions 1800 apart and gradually decreases as GTP is rotated and is completely zero at 900 away. To detect circularly polarized light, we again use GTP . Since the amplitude of circularly polarized light is same in all direction, we see no change in intensity of

52 emergent beam through GTP in a full rotation.

53 Appendix B

Calculation of Circular Polarization Degree

The polarization degree of any atomic transition can be calculated by using density matrix elements. The general form of the density matrix element is given as,

0 0 k 0 0 0 k hj m | Tq | jmi = C(jkj ; mqm ) hj k T k ji (B.1) where C(j k j’ ; m q m’) is Clebsch-Gordan coefficient. It follows the conservation of angular momentum and vanishes unless m0 = q+m. The second term hj0 k T k k ji k is called the reduced matrix elements of tensor operator Tq and is independent of quantum numbers m and m0 . j0 ,m0 are the quantum numbers for final state and j, m are the quantum numbers for initial state. k is the rank of tensor operator and q is total quantum number of the transition. 2 2 Here is the calculation of circular polarization degree for 6s S1/2→10s S1/2 The transitions (A) and (B) from the ground state to intermediate state can be

Figure B.1: Transition of electrons by right circularly polarized light. represented by the following density matrices.

54 For transition (A), ¿3 1 1 1À hj m | r | j m i = | r | − 2 2 q 1 1 2 2 +1 2 2 1 3 1 1 ¿3 1À = C( 1; − 1) k T k k 2 2 2 2 2 2 (B.2) For transition (B), ¿3 3 1 1À hj m | r | j m i = | r | 2 2 q 1 1 2 2 +1 2 2 1 3 1 3 ¿3 1À = C( 1; 1) k T k k 2 2 2 2 2 2 (B.3) Now, Clebsch-Gordan coefficients for given matrices are, For transition A,

v u u 1 0 3 t j1 + m + 2 1 a1(m ) = −(j1 − 3m + ) = 2 2j1(2j1 + 1)(2j1 + 3) 2 and, For transition B,

v u √ u 1 1 3 0 t3(j1 + m − 2 )(j1 + m + 2 )(j1 − m + 2 ) 3 a2(m ) = − = − 2j1(2j1 + 1)(2j1 + 3) 2 Now alignment is,

0 2 02 X 0 2 02 0 0 h3Jz − J i |a(m )| [3m − J (J + 1)] 2 hA0i = 0 0 = 0 0 = J (J + 1) m0 J (J + 1) 5 and orientation is ,

0 X 0 2 0 hJzi |a(m )| [m ] 5 hOoi = q = q hOoi = √ 0 0 0 0 J (J + 1) m0 J (J + 1) (2 15) Using Equation 3.24, the calculated polarization degree is 100%. This is the polarization without hyperfine depolarization effect. However if the depolarization effect is considered then the polarization degree is different and is given by,

√ (1) 15hOoig PC = (2) 2 + 5/4hAoig (1) (2) PC = 59.8% where g = 0.4764 and g = - 0.0892

55 133 2 2 Table B.1: The alignment and orientation in the Cs 6p P1/2 and 6p P3/2 states.

Circular polarization (σ+) Circular polarization (σ−) Level hA i hO i P (%) hA i hO i P (%) o √o C o √o C P1/2 0 1/√3 100 0 -1/√3 -100 P3/2 2/5 5/2 15 100 2/5 -5/2 15 -100

Table B.2: Clebsch-Gordan coefficients used in the experiment

C( 3 1 1; −1 1 1) −1 2 2 2 2 q2 3 1 −3 −1 3 C( 2 2 1; 2 2 1) 4 C( 1 3 1; − −1 1 1) 1 2 2 2 2 q2 1 3 1 3 3 C( 2 2 1; 2 2 1) − 4

56 Appendix C

LabVIEW Program

Figure C.1: Interface of the LCVR.

57 Figure C.2: Medowlark LCR subVI.

Figure C.3: takedata3 sub.vi. ¯

58 Figure C.4: Stepper motor subVI.

Figure C.5: Get Spectrum program to take wave form of fluorescence signal.

59 Appendix D

Apparatus

Figure D.1: Experimental apparatus.

60 Figure D.2: Flowing dye laser 1 oscillator in the Littman-Metcalf design.

Figure D.3: Dye flowing machine for laser 1.

61 Figure D.4: Static dye laser 2 oscillator in Littman-Metcalf cavity.

Figure D.5: Dye laser 2 amplifier.

62 Figure D.6: A view of quarter wave plate, Glan-Thompson polarizer and the LCR .

Figure D.7: Boxcar averager/integrator.

63 Figure D.8: DAQ board connected between a computer and the boxcar aver- ager/integrator.

64